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Error Mitigation with Quantum Subspace Expansion

João Carlos de Andrade Getelina

Funding:

arXiv:2404.09132

(soon to be published APL Quantum)

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Collaborators

Prachi Sharma

Peter Orth

Tom Iadecola

Yongxin Yao

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Outline

  • Introduction:
    • Motivation and background to subspace methods;
    • Quantum subspace expansion;
  • Methodology:
    • Integrating VQE with QSE;
    • Models and methods;
  • Results and discussion:
    • Our main findings;
    • What’s next?

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Motivation

  • Ground-state preparation (NISQ): VQE and QITE;
  • Increasing acc. circuit depth (noiseless);
  • Mitigation techniques (ZNE and PEC) depth;
  • Improving acc. with the same overhead cost?

Quantum Subspace Expansion

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Krylov subspace methods

  • General outlook:

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  • Finding (a few) useful eigenpairs of matrices;
  • Ex: Arnoldi iteration (general) and Lanczos (Hermitian);
  • Requires orthogonalization schemes;

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Quantum subspace expansion

  • Applying the same idea of KSE in a quantum setup:

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  • Problem: QPUs only give us bitstring counts;
  • Generalized eigenvalue problem:

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Quantum subspace expansion

  • QLanczos: [Nat. Phys. 16, 205 (2020)]

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  • Unsuitable for NISQ (Hadamard test);
  • FGKS: taking each term of as a subspace vector;
  • Amounts to exp. val. of Pauli strings;
  • Goal: balance CPU and QPU cost;

[Phys. Rev. A 104, L050401 (2021)]

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Outline

  • Introduction:
    • Motivation and background to subspace methods;
    • Quantum subspace expansion;
  • Methodology:
    • Integrating VQE with QSE;
    • Models and methods;
  • Results and discussion:
    • Our main findings;
    • What’s next?

​

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Variational quantum eigensolver

  • Variational principle of QM:
  • HVA: apply unitaries like Hamiltonian terms;
  • Considering simple 1D and 2D spin models:

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VQE under HVA

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Accuracy of noisy VQE simulations

  • How does one improve accuracy with VQE?

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Adding layers

More noise

Worst acc.

  • Improving HVA acc. w/o increasing quantum overhead?

QSE: more measurements of the same circuit

  • Key parameters: circuit depth and expansion order

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Pairing VQE and QSE

  1. CPU: Find optimal parameters under HVA;
  2. QPU: Build circuit based on step 1;
  3. QPU: Measure Pauli strings of the FGKS;
  4. CPU: Get and and solve GEP;

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Pairing VQE and QSE

  • FGKS ex: 3-site 1D TFIM

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Pairing VQE and QSE

  • Subspace basis transformation:
  • Eigenbasis of overlap matrix:
  • Get corresponding GS:

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Outline

  • Introduction:
    • Motivation and background to subspace methods;
    • Quantum subspace expansion;
  • Methodology:
    • Integrating VQE with QSE;
    • Models and methods;
  • Results and discussion:
    • Our main findings;
    • What’s next?

​

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Pauli groups scaling

  • Ex: 2D TFIM;
  • Greedy alg.: largest degree first

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Fidelity and energy convergence

Compare KS with FGKS:

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Shot noise sims.

  • Proxying quality of expanded state (recall):

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  • GEP no longer bounded;
  • Sing. val. cutoff: biased;
  • Trace criterion (unbiased);
  • L=1, K=2, 4x4 TFIM

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Pairing VQE and QSE

  • Subspace basis transformation:
  • Eigenbasis of overlap matrix:
  • Get corresponding GS:

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5-qubit MFIM

  • Opt. lvl 1 vs. 3;
  • Ent. gate parameter:

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  • QEM: TREX

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16-qubit MFIM

  • L=1, K=1
  • 14672 Pauli strings
  • 83 self-commuting groups
  • QEM: ZNE + DD + PT + TREX
  • Evaluate w.r.t. noise-scaled and

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Overview

  • VQE + QSE improves acc. with little cost;
  • Suitable method for NISQ;
  • Larger systems: requires combining other QEMs;
  • Ref. state computed only classically so far;

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What’s next?

  • Noisy ref. state;
  • Co-optimized VQE;
  • Ranking Pauli groups;
  • Testing other QEMs;
  • Trying different ref. states (e.g., QITE);